Instead of diagonalizing the Hamiltonian directly, we use the gradient descent iteration as in the FQE approach [1]} using the LCU formalism [2]} in QCSH. We start from a trial wavefunction \(|\phi ^{(t)}\rangle \) , and the average energy of the system is \(\langle E\rangle = \langle \phi ^{(t)}|H|\phi ^{(t)}\rangle \) . Taking the gradient of the average energy, we have \(\nabla \langle \phi ^{(t)}|H|\phi ^{(t)}\rangle = 2 H |\phi ^{(t)}\rangle \) . The next step will be iterated as
\(|\phi ^{(t+1)}\rangle \propto (I-2\gamma H)|\phi ^{(t)}\rangle ,\) 
